Chapter 6: Problem 74
Multiple Choice \(\int_{0}^{2} e^{2 x} d x=\) (A) \(\frac{e^{4}}{2} \quad(\mathbf{B}) e^{4}-1 \quad\) (C) \(e^{4}-2 \quad\) (D) \(2 e^{4}-2 \quad(\mathbf{E}) \frac{e^{4}-1}{2}\)
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In Exercises 67 and \(68,\) make a substitution \(u=\cdots(\) an expression in \(x), \quad d u=\cdots .\) Then (a) integrate with respect to \(u\) from \(u(a)\) to \(u(b)\) . (b) find an antiderivative with respect to \(u,\) replace \(u\) by the expression in \(x,\) then evaluate from \(a\) to \(b\) . $$\int_{\pi / 6}^{\pi / 3}(1-\cos 3 x) \sin 3 x d x$$
In Exercises 67 and \(68,\) make a substitution \(u=\cdots(\) an expression in \(x), \quad d u=\cdots .\) Then (a) integrate with respect to \(u\) from \(u(a)\) to \(u(b)\) . (b) find an antiderivative with respect to \(u,\) replace \(u\) by the expression in \(x,\) then evaluate from \(a\) to \(b\) . $$\int_{0}^{1} \frac{x^{3}}{\sqrt{x^{4}+9}} d x$$
True or False If \(f\) is positive and differentiable on \([a, b],\) then $$\int_{a}^{b} \frac{f^{\prime}(x) d x}{f(x)}=\ln \left(\frac{f(b)}{f(a)}\right) .$$ Justify your answer.
In Exercises \(17-24,\) use the indicated substitution to evaluate the integral. Confirm your answer by differentiation. $$\int 8\left(y^{4}+4 y^{2}+1\right)^{2}\left(y^{3}+2 y\right) d y, \quad u=y^{4}+4 y^{2}+1$$
Extinct Populations One theory states that if the size of a population falls
below a minimum \(m,\) the population will become extinct. This condition leads
to the extended logistic
differential equation \(\frac{d P}{d t}=k
P\left(1-\frac{P}{M}\right)\left(1-\frac{m}{P}\right)\)
with \(k>0\) the proportionality constant and \(M\) the population maximum.
(a) Show that dP&dt is positive for m < P < M and negative if P
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